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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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-1 votes
0 answers
77 views

So I know that $\lim_{n\to\infty} \ln(n)=\infty$; I've seen some proof online using the mean value theorem. But is it not easier to assume that it converges, so that $\lim_{n\to\infty} \ln(n)=k$ where ...
3 votes
2 answers
120 views

How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n,ドル where $n$ is a natural number? I discovered this identity while trying to prove Prove using ...
2 votes
0 answers
56 views

Consider any matrix $A \in \text{GL}_d(\mathbb{C}),ドル i.e, a square invertible matrix. We define a logarithm of $A$ as any matrix $X$ such that $$e^X = A.$$ Our objective is to find of possible ...
8 votes
3 answers
623 views

There is something about the branch of complex logarithms that I do not understand correctly... Any clarification would be appreciated! Let $f(z) = z^2$ and let $\Omega = \mathbb{C} \setminus \mathbb{...
0 votes
1 answer
72 views

I have a problem related to the idle game Clicker Heroes, in finding the expected $\log_{10}$ value of the rewards gained from defeating bosses throughout an ascension. For boss number $k,ドル the reward ...
0 votes
1 answer
100 views

I have a problem stemming from looking into ways to approximate the Lambert W function on a graphing program like Desmos. In my process of graphing functions, I came across a question I never had ...
-4 votes
2 answers
246 views

I am refreshing my calculus knowledge using. The workbook direction for the problem is: Perform the following derivative, where: $$\cot(4 \theta^2 - 1) > 0.$$ The author then presents the below ...
0 votes
0 answers
40 views

I am trying to solve an optimization problem that contains power functions. I reformulated the problem via a logarithmic function, and it works well. The terms of the problem involved are similar to $\...
-3 votes
1 answer
72 views

I see the proccedure of: How I find the limit of $\frac{2^n}{e^{p(n)l}}$ I didn’t understand how it applies in my case: $$ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$$ $$ \lim_{x\to\infty} \frac{\...
2 votes
2 answers
219 views

I cannot find a closed form solution for $x$ in $\dfrac{x^2 e^x}{e^x - 1} = k$ where $\{x,k \} \in \mathbb{R}^+$. I thought there might be a PolyLog solution, but apparently there isn't. Is there ...
11 votes
4 answers
498 views

I am trying to prove the subsequent statement, but I did not make any progress as of yet. $$\lim_{n\:\!\to\:\!\infty} ,円\sum_{k=1}^{n-1} \frac{1}{2^n}\binom{n}{k}\log_2\!\left(\frac{n}{k}\right) = 1$$ ...
1 vote
2 answers
264 views

Determine the smallest possible value of the natural number $ a_1,ドル knowing that there exist natural numbers $ a_1 \geq a_2 \geq \ldots \geq a_{100} \geq 2 $ with the property that $$ \left\{ \sum_{k=...
2 votes
1 answer
131 views

I understand that $\log_1(1)$ is considered an indeterminate form, but the expression $\log_0(0)$ seems even more subtle. Algebraically, it is undefined because a logarithm cannot have a base of zero, ...
0 votes
1 answer
123 views

We have the following inequality for the logarithm: given any 0ドル<a<1,ドル there exists a constant $C_a$ such that $$\log (1+x) \leq C_a \frac{x}{(1+x)^a} $$ holds for all $x>0$. In other words, ...
2 votes
1 answer
215 views

I have seen a lot of references on the topic but none of them really contained a proof from first principles that the monodromy group of logarithm is $\mathbb{Z}$. Here's the developed theory in my ...

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